Download PDF by Ivor Grattan-Guinness: Companion encyclopedia of the history and philosophy of the

By Ivor Grattan-Guinness

ISBN-10: 0415092396

ISBN-13: 9780415092395

Arithmetic is likely one of the most simple -- and so much old -- varieties of wisdom. but the main points of its ancient improvement stay vague to all yet a number of experts. The two-volume spouse Encyclopedia of the background and Philosophy of the Mathematical Sciences recovers this mathematical historical past, bringing jointly a number of the world's prime historians of arithmetic to check the background and philosophy of the mathematical sciences in a cultural context, tracing their evolution from precedent days to the 20th century.In 176 concise articles divided into twelve elements, individuals describe and study the range of difficulties, theories, proofs, and strategies in all parts of natural and utilized arithmetic, together with likelihood and statistics. This fundamental reference paintings demonstrates the ongoing value of arithmetic and its use in physics, astronomy, engineering, computing device technology, philosophy, and the social sciences. additionally addressed is the historical past of upper schooling in arithmetic. conscientiously illustrated, with annotated bibliographies of assets for every article, The better half Encyclopedia is a worthwhile learn device for college kids and academics in all branches of mathematics.Contents of quantity 1: -Ancient and Non-Western Traditions -The Western center a long time and the Renaissance -Calculus and Mathematical research -Functions, sequence, and strategies in research -Logic, Set Theories, and the principles of arithmetic -Algebras and quantity TheoryContents of quantity 2: -Geometries and Topology -Mechanics and Mechanical Engineering -Physics, Mathematical Physics, and electric Engineering -Probability, records, and the Social Sciences -Higher schooling andInstitutions -Mathematics and tradition -Select Bibliography, Chronology, Biographical Notes, and Index

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Additional resources for Companion encyclopedia of the history and philosophy of the mathematical sciences, Volume 2

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His success in this matter, of which he was very proud, was one of the reasons for his confidence in the general efficacy of the new approach. Figure 4 Pappus’s problem However, all was not so simple. In Descartes’s time, a solution to a problem in geometry had naturally to be given in geometric terms. Bos shows in two important papers (1981, 1984), this historical circumstance had a determining effect on how Descartes presented his ideas. For Pappus’s problem, it was possible to describe the answer in geometrical terms: the solution curve is a conic section.

In modern notation, such a polygon would be designated {m/h}; convex polygons correspond to h=1, and the pentagram {5/2} to m=5 and h=2 (see Figure 4, taken from Poinsot’s paper). The last row in Figure 4 illustrates (for m=6 and h=2) Poinsot’s (correct) statement that this construction does not produce a regular polygon if m and h are not relatively prime; in such a case the resulting figure is composed of several regular polygons, and hence is not itself a regular polygon. The logical error committed by Poinsot and uncritically accepted by all later writers is the assumption that this reasoning proves the non-existence of regular polygons (as defined) other than the ones corresponding to relatively prime m and h.

With a=1/2, the trisectrix gives a length of 1/π, which can easily be made to yield a length of π, thus squaring the circle. For this reason the trisectrix is also called the ‘quadratrix’ (it has brought about the quadrature of the circle). At a time when it had to be shown rigorously that the concept of area, first introduced for rectilinear figures, applies to the circle, this was a considerable achievement. Duplicating the cube recalls the problem of duplicating the square, but whereas the latter problem only requires √2 to be found, and was solved in Plato’s dialogue the Meno, duplicating the cube requires ∛2 to be found, which is much harder.

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Companion encyclopedia of the history and philosophy of the mathematical sciences, Volume 2 by Ivor Grattan-Guinness

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